Double Dirichlet series associated with arithmetic functions

Abstract

This paper is a continuation of our previous work on double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the Möbius function, and so on. We consider the analytic behaviour around the non-positive integer points on singularity sets which are points of indeterminacy. In particular, we show a certain reciprocity law of their residues. Also on this occasion we correct some inaccuracies in our previous paper.

A survey on the theory of multiple Dirichlet series with arithmetical coefficients on the numerators

“…. A recent article of A. Nawashiro, Tsumura and the author [29] studied several examples which satisfy N(Φ x…”

“…The paper [29] only considers the double zeta case, but a generalization of [29] to the general multiple case was treated by Rei Kawashima [24]. She also discussed the case when Λ is replaced by the Liouville function λ.…”